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Plane whose normal is along a given segment (perpendicular bisector or perpendicular to a joining line)

This type of question, asked 5 times: 2021 Paper 9, 2021 Paper 25, 14 Jun 2022, 31 Jan 2023, 1 Feb 2023. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 9

    82
    Let the plane ax+by+cz+d=0ax+by+cz+d=0 bisect the line joining the points (4,−3,1)(4,-3,1) and (2,3,−5)(2,3,-5) at right angles. If a,b,c,da,b,c,d are integers, then the minimum value of (a2+b2+c2+d2)\left(a^{2}+b^{2}+c^{2}+d^{2}\right) is

  2. Question 2

    2021 Paper 25

    87
    Let (λ,2,1)(\lambda,2,1) be a point on the plane which passes through the point (4,−2,2)(4, - 2,2). If the plane is perpendicular to the line joining the point (−2,−21,29)( - 2, - 21,29) and (−1,−16,23)( - 1, - 16,23), then (λ11)2−4λ11−4\left( \frac{\lambda}{11} \right)^{2}-\frac{4\lambda}{11}- 4 is equal to

  3. Question 3

    14 Jun 2022

    165
    Let the points on the plane PP be equidistant from the points (−4,2,1)( - 4,2,1) and (2,−2,3)(2, - 2,3). Then the acute angle between the plane PP and the plane 2x+y+2x + y + 3z=13z = 1 is

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  4. Question 4

    31 Jan 2023

    166
    Let PP be the plane, passing through the point (1,−1,−5)(1, - 1, - 5) and perpendicular to the line joining the points (4,1,−3)(4,1, - 3) and (2,4,3)(2,4,3). Then the distance of PP from the point (3,−2,2)(3, - 2,2) is

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  5. Question 5

    1 Feb 2023

    171
    Let αx+βy+γz=1\alpha x + \beta y + \gamma z = 1 be the equation of a plane through the point (3,−2,5)(3, - 2,5) and perpendicular to the line joining the points (1,2,3)(1,2,3) and (−2,3,5)( - 2,3,5). Then the value of αβγ\alpha\beta\gamma is equal to